Ntru Cryptography: A Tutorial · 5/23/2006 [email protected] 1 Ntru Cryptography: A Tutorial Wei Ren, Ph.D Department of Electrical and Computer Engineering University of Nevada, Las

5/23/2006 [email protected] 1

Ntru Cryptography: A Tutorial

Wei Ren, Ph.DDepartment of Electrical and Computer EngineeringUniversity of Nevada, Las Vegas

Email: [email protected] 23, 2006

http://www.cs.unlv.edu/~renw/ntru-tutorial-slides.pdf

[email protected] 25/23/2006

AgendaAlgebra Tutorial

Modular ArithmeticTruncated Polynomial RingsInverse in Truncated Polynomial Rings

The NTRU Public Key CryptosystemNTRU PKCS ParametersKey GenerationEncryptionDecryptionWhy It Works

Advanced Topics (Optimizations)Implementation Details


Presentation OutlineAlgebra Tutorial





Modular Arithmetic

Division with modulo and keep the remaindere.g. 147 (modulo 17) =?

147=8*17+11 that is 147=11 (modulo 17)In general, the congruence a=b (modulo m) means that a and b leave the same remainder when they are divided by m.(a modulo m) + (b modulo m)=(a+b modulo m)(a modulo m) * (b modulo m)=(a*b modulo m)If a*b=1 (modulo m), b is an inverse for a (modulo m)e.g. inverse of 10 (modulo 23) is 7, why?

7*10=1(modulo 23)Euclidean Algorithm can be used to check if a and m have common factors and compute the inverse of a (modulo m) if they do not have common factors


Truncated Polynomial RingsDegree N-1 RingE.g. a =a0+a1X+a2X2+a3X3+…+aN-2XN-2+aN-1XN-1

a+b=(a0+b0)+(a1+b1)X+…(aN-1+bN-1)XN-1

XN=1 (mod XN -1)a*b=c0+c1X+c2X2+…+cN-2XN-2+cN-1XN-1

ck=a0bk+a1bk-1+…+akb0+ak+1bN-1+ak+2bN-2+…aN-1bk+1

a*(b+c)=a*b+a*cCall it Ring of Truncated Polynomials. In terms of modern abstract algebra, R is isomorphic to the quotient ring Z[X]/(XN-1)

∑∑∑≡+

−

+=

−+

=

− =+=)(mod

1

10 Nkji

ji

N

ki

ikNi

k

i

ikik bababac


Truncated Polynomial R with the modular arithmetic

Polynomial a modulo an integer qa (modulo q)Means to reduce the coefficients of a modulo q

a=b (modulo q)Means every coefficients of the difference a-b is a multiple of q

a =a0+a1X+a2X2+a3X3+…+aN-2XN-2+aN-1XN-1

is conveniently written as the list of N numbersa=(a0, a1, a2, …..,,aN-2, aN-1)

e.g. when N=7, polynomial a = 3+2X2-3X4+X6 is stored as the list (3,0,2,0,-3,0,1)


Inverses in Truncated Polynomial R

Inverse modulo q of a polynomial a modulo is a polynomial A withthe property that

a*A=1 (modulo q)Not every polynomial has an inverse modulo q, but it is easy to determine if a has an inverse and to compute the inverse if it exists

e.g. N=7, q=11, a = 3+2X2-3X4+X6, the inverse of a modulo 11 is A=-2+4X+2X2+4X3-4X4+2X5-2X6

Since(3+2X2-3X4+X6)*(-2+4X+2X2+4X3-4X4+2X5-2X6)= -10+22X+22X3-22X6

=1 (modulo 11)







NTRU PKCS ParametersRing R that consists of all truncated polynomials of degree N-1 having integer coefficients:

a =a0+a1X+a2X2+a3X3+…+aN-2XN-2+aN-1XN-1

N: the polynomials in the truncated polynomial ring have degree N-1q: large modular, the coefficients of the truncated polynomials will be reduced mod qp: small modular, as the final step in decryption, the coefficients of the message are reduced mod p


NTRU PKCS Parameters

3256503Highest3128347High3128251Standard 3128167Moderate

pqNSecurity Level

Ntru167 ECC112 RSA512Ntru263 ECC168 RSA1024Ntru503 ECC196 RSA2048

From www.ntru.com, ntru tutorial

In this tutorial, N=11, q=32, p=3


Key Generation

Randomly Choose two “small” polynomials f and g and keep them privateRandomly means coefficients is randomly distributed in p or q, small means the coefficients are much smaller than p or qCompute the inverse of f modulo q and the inverse of f modulo p

f*fq=1 (modulo q) and f*fp=1 (modulo p)

Public Key is: h=pfq*g (modulo q)


Key Generation ExampleN=11, q=32, p=3Some method to generate f and g:df: The polynomial f has df coefficients equal to +1 and df -1 coefficients equal to -1, and all the rest are 0dg : The polynomial g has dg coefficients equal to +1 and dg coefficients equal to -1, and all the rest are 0The reason: f and g are “small” polynomials, f has to be inverse while g doesn’tdf=4 dg=3

f=-1+X+X2-X4+X6+X9-X10

g=-1+X2+X3+X5-X8-X10


Key Generation Example (cont.)

fp=1+2X+2X3+2X4+X5+2X7+X8+2X9

fq=5+9X+6X2+16X3+4X4+15X6+22X7+20X8+18X9+30X10

How to generate fp and fq? Discuss it later.

H=pfq*g (modulo q) q=32, p=3g=-1+X2+X3+X5-X8-X10 (in previous slide)

H=8+25X+22X2+20X3+12X4+24X5+15X6+19X7+12X8+19X9+16X10


Is fp really inverse of f , fp*f=1 (mod p) p=3 ? Verification

XN=1 (mod XN -1)a*b=c0+c1X+c2X2+…+cN-2XN-2+cN-1XN-1

ck=a0bk+a1bk-1+…+akb0+ak+1bN-1+ak+2bN-2+…aN-1bk+1

fp=1+2X+2X3+2X4+X5+2X7+X8+2X9

f = -1+X+X2-X4+X6+X9-X10

e. g. c0=1*(-1)+2*(-1)+0*1+2*0+2*0+1*1+

0*0+2*(-1)+1*0+2*1+0*1=(-1)+(-2)+1+(-2)+2= -2

Since p=3(-2) =1 (mod 3)

∑∑∑≡+

−

+=

−+

=

− =+=)(mod

1

10 Nkji

ji

N

ki

ikNi

k

i

ikik bababac

(1, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0) (-1, 1, 1, 0, -1, 0, 1, 0, 0, 1, -1)


How to compute H=pfq*g (mod q) q=32, p=3

Low Hamming Weight PolynomialsReference: J.Hoffstein, J.Silverman, “Random Small Hamming Weight Products With Applications to Cryptography,”http://www.ntru.com/cryptolab/articles.htm, Last Access, May 19,2006

e.g. (4,5,7)*(5,3,2)=4*(5,3,2)+5*(2,5,3)+7*(3,2,5)=(20,12,8)+(10,25,15)+(21,14,35)=(20+10+21, 12+25+14, 8+15+35)= (51,51,56)


How to compute H=pfq*g (mod q) q=32, p=3 (cont.)

Using Low Hamming Weight Polynomialsfq=5+9X+6X2+16X3+4X4+15X5+16X6+22X7+20X8+18X9+30X10

g=-1+X2+X3+X5-X8-X10

H=pfq*g (mod q) p=3, q=32

(-1,0,1,1,0,1,0,0,-1,0,-1)*(5,9,6,16,4,15,16,22,20,18,30)=(-5,-9,-6,-16,-4,-15,-16,-22,-20,-18,-30)+(18,30,5,9,6,16,4,15,16,22,20)+(20,18,30,5,9,6,16,4,15,16, 22)+(16,22,20,18,30,5,9,6,16,4,15)+(-16,-4,-15,-16,-22,-20,-18,-30,-5,-9)+(-9,-6-16,-4,0,-15,-16,-22,-20,-18,-30,-5)=(24,51….)

-5+18+20+16-16-9=24 24*3=72 72=8 (mod 32)

H=8+25X+22X2+20X3+12X4+24X5+15X6+19X7+12X8+19X9+16X10


Encryption

m is plaintext in the form of a polynomial whose coefficients are “small” mod qRandomly choose another “small” polynomial rr is “blinding value” which is used to obscure the message (similar to the way that ElGamalalgorithm use a one-time random value when encrypting)e = r*h +m (modulo q), e is encrypted message, m is plaintext, h is public key


Encryption Example

r has dr coefficients equal to 1, dr-1 coefficients equal to -1, and all others are 0dr=3, r=-1+X2+X3+X4-X5-X7

m=-1+X3-X4-X8+X9+X10

h=8+25X+22X2+20X3+12X4+24X5+15X6+19X7+12X8+19X9+16X10

e=r*h+m (mod q)=(-1, 0,1,1,1,-1,0,-1,0,0,0)*(8,25,22,20,12,24,15,19,12,19,16)+(-1,0,0,1,-1,0,0,0,-1,1,1)

=14+11X+26X2+24X3+14X4+16X5+30X6+7X7+25X8+6X9+19X10


Decryption

a = f*e (mod q), MUST choose coefficients of a to lie between -q/2 and q/2, e.g. for q=32, coefficients must lie in [-15, 16]b = a (mod p), MUST choose coefficients of b between -p/2 and p/2, for p=3, the range is [-1,1]c = fp*b (mod p), MUST choose coefficients of c between -p/2 and p/2, for p=3, the range is [-1,1]


Decryption Example: a=f*e (mod q)

e=14+11X+26X2+24X3+14X4+16X5+30X6+7X7+25X8+6X9+19X10

f = -1+X+X2-X4+X6+X9-X10

(-1,1,1,0,-1,0,1,0,0,1,-1)*(14,11,26,24,14,16,30,7,25,6,19)mod 32, change coefficients to [-15,16]a=3-7X-10X2-11X3+10X4+7X5+6X6+7X7+5X8-3X9-7X10

denoted by (3,-7,-10,-11,10,7,6,7,5,-3,-7)


Decryption Example: b=a (mod p)

a=3-7X-10X2-11X3+10X4+7X5+6X6+7X7+5X8-3X9-7X10

(3,-7,-10,-11,10,7,6,7,5,-3,-7)

b=a (mod 3), change coefficients to [-1,1]

b=-X-X2+X3+X4+X5+X7-X8-X10 (mod 3)(0,-1,-1,1,1,1,0,1,-1,0,-1)


Decryption Example: c=fp*b (mod p)

fp=1+2X+2X3+2X4+X5+2X7+X8+2X9

(1,2,0,2,2,1,0,2,1,2,0)b=-X-X2+X3+X4+X5+X7-X8-X10

(0,-1,-1,1,1,1,0,1,-1,0,-1)

(0,-1,-1,1,1,1,0,1,-1,0,-1)*(1,2,0,2,2,1,0,2,1,2,0)=(0,-1,-2,0,-2, -2,-1,0,-2,-1,-2,0)+(0,0,-1,-2,0,-2, -2,-1,0,-2,-1,-2)+(1,2,0,1,2,0,2,2,1,0,2)+ (2,1,2,0,1,2,0,2,2,1,0)+(0,2,1,2,0,1,2,0,2,2,1)+(2,1,0,2,1,2,0,1,2,0,2)+(-2,-2,-1,0,-2,-1,-2,0,-1,-2,0)+(-2,0,-2,-2,-1,0,-2,-1,-2,0,-1)mod 3, change to [-1,1], therefore c = (-1,0,0,1,-1,0,0,0,-1,1,1)

m = (-1,0,0,1,-1,0,0,0,-1,1,1)

equal


SummaryParameters: N, p (small prime), q (big number, power of 2, gcd(p,q)=1)Private Key: Two randomly generated “small”polynomials f, gComputing fq, fp, fq*q=1 (mod q), fp*p=1(mod p)Public key: h=pfq*g (mod q)Encryption: randomly generated “small” polynomial r as blind valuee=r*h+m (mod q), e is cipher text, m is plaintextDecryption: a=f*e (mod q), b=a (mod p), c=fp*b (mod p), change the coefficients, c is the result, which should be equal to m


Why it Works

a=f*e (mod q) =f*(r*h+m) (mod q) =f*(r*pfq*g+m) (mod q) =pr*g+f*m (mod q)

[e=r*h+m (mod q)]

[h=pfq*g (mod q)]

[f*fq=1 (mod q)]

The polynomial r, g, f, m all have coefficients that are quite small, so the coefficients of r*g and f*m are also quite small, at least in comparison to q. Since prime p is also small compared to q, this means the polynomial pr*g+f*m lie between –q/2 and q/2, so reducing the coefficients mod q has no effect.

b=a=f*m (mod p)c=fp*b=fp*f*m=m (mod p) [since fp*f=1 (mod p)]







Optimizations

Reference:J. Hoffstein, J. Silverman, “Optimizations for NTRU,” In: Proc. of Public-Key Cryptography and Computational Number Theory (Warsaw, September 11-15, 2000), Walter de Gruyter, Berlin-New York, 2001.J. Hoffstein, J. Silverman, “Random Small Hamming Weight Products with Applications to Cryptography,” In: Proc. of Com2MaC Workshop on Cryptography(Pohang, Korea, June 2000), Discrete Mathematics, to appear.


Optimizations (cont.)

Polynomial MultiplicationLow Hamming Weight Polynomialse.g. m, f, g, r, Products of Small Hamming Weight Polynomialse.g. h=pfq*g (mod q), e=r*h+m (mod q), a=f*e (mod q)Instead of taking f to be a single small polynomials, form it by combining several even smaller polynomials

e.g. in full-size versions of the cryptosystem, with N=251, usually take small polynomials so that about one third of the coefficients are non-zero



e.g. for computing i*a, i=[1,0,1,1,1,1,1,1,0,0,1,1,0]i=i1*i2=[1,0,1,0,0,0,0,1,0,0,0,0,0]*

[1,0,0,1,1,0,0,0,0,0,0,0,0]I has 9 ones, so i*a take 9 additions pre coefficientIf we instead I with i1 and i2 First calculate i2*a, it take 3 additions pre coefficient, then calculate i1*(i2*a), it take another 3 additions per coefficient, so the total is 6 additions per coefficient, only take 2/3 as long



For commercial ntru, N=251Short vector have about 72 non-zero coefficientsFor decryption, a = f*e (mod q), let f=1+p*FdF=72, f=1+p*((f1*f2)+f3)df1=8,df2=8,df3=8, so it takes 24 additions pre coefficients, not 72For encryption, e=r*h+m (mod q), let r=(r1*r2)+r3 dr1=8, dr2=8, dr3=8, so it takes 24 additions pre coefficients, not 72







Implementation Details

Source code:www.cs.unlv.edu/~renw/ntru_v22.c

Document:www.cs.unlv.edu/~renw/ntru-tutorial-impl.pdf

Language: ANSI CCompile: gcc ntru_v22.c –o ntruUsage: ntru plaintext (max length is 11, ‘0’ and ‘1’ character)eg. Ntru 11111000001


Functions in Program

GF_Ntru_ParameterSetup(11,32,3);GF_Ntru_PrivateKeyGen();GF_Ntru_PublicKeyGen();GF_Ntru_BlindValueGen();GF_Ntru_GetPlainText();GF_Ntru_Encrypt();GF_Ntru_Decrypt();GF_Debug_Check_Result();


Program Organizations

GF_Ntru_ParameterSetup(11,32,3)

GF_Ntru_PrivateKeyGen()

GF_Ntru_PublicKeyGen()

GF_Ntru_BlindValueGen()

GF_Ntru_GetPlainText()

GF_Debug_Check_Result()

GF_Ntru_Encrypt()

GF_Ntru_Decrypt()

Main()

Data Flow Diagram Function-Calling Graph

GF_Ntru_ParameterSetup

GF_Ntru_PrivateKeyGen

GF_Ntru_PublicKeyGen

GF_Ntru_BlindValueGen

GF_Ntru_GetPlainText

GF_Ntru_Encrypt

GF_Ntru_Decrypt


Implementation Details (cont.)for(t=0;t<GV_N;t++) //using low weight hamming polynomial multiplication{

if (r[t]==1){

for(i=0;i<GV_N;i++)e[i]=e[i]+h[i];

}if (r[t]==-1){

for(i=0;i<GV_N;i++)e[i]=e[i]-h[i];

}// h[ ] one right shiftint swaptemp=h[GV_N-1];for (i=GV_N-1;i>0;i--){

h[i]=h[i-1];}h[0]=swaptemp;

}

Low weight hamming polynomial product, e.g. e=r*h + m (mod q)


What’s Next

Implement commercial version, NTRU503, (on-going, in debugging stage)Hardware-software co-design Optimization

montogomery multiplication hardware implementation (VHDL, ModelSim)

Performance comparison between RSA, ECC in sensor network platformScrutiny of NTRU security (Lattices) Ntru-based Key management (authentication, signature) for wireless sensor network security


Acknowledgement

I would like to thank Prof. Yoohwan Kim, Prof. Mei Yang and Prof. Yingtao Jiang for the insightful comments and discussions.


ReferencesThe NTRU Public Key Cryptosystem – A Tutorial,

http://www.ntru.com/cryptolab/tutorials.htm, last access is May 19, 2006

The End and Thanks

Ntru Cryptography: A Tutorial · 5/23/2006 [email protected] 1 Ntru Cryptography: A Tutorial Wei Ren, Ph.D Department of Electrical and Computer Engineering University of Nevada, Las

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